[Final] A Theory of Identity and Consumption

The Market for Apples:

A Theory of Identity and Consumption Clement Lee

Under the supervision of

Dr. Rachel E. Kranton

Dr. Michelle P. Connolly

Department of Economics, Duke University

Durham, North Carolina

2016

Honor thesis submitted in partial fulfillment of the requirements for Graduation with

Distinction in Economics in Trinity College of Duke University

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Table of Contents

Acknowledgement .............................................................................................................. 3

Abstract ............................................................................................................................... 4

1. Introduction ..................................................................................................................... 5

2. Literature Review ............................................................................................................ 8

3. Basic Model .................................................................................................................. 10

4. Market Demand ............................................................................................................ 15

4.1. Model Specification ............................................................................................... 15

4.2. Interpretation of Shape Variables .......................................................................... 16

4.3. Solving the Model .................................................................................................. 18

4.4. Interaction between Identity and Market Demand ................................................. 21

5. Salience ......................................................................................................................... 24

6. Chick-fil-A .................................................................................................................... 27

7. Conclusion .................................................................................................................... 31

References ......................................................................................................................... 32

Appendix ........................................................................................................................... 35

Proof to Lemma 1 ......................................................................................................... 35

Proof to Lemma 2 ......................................................................................................... 36

Proof to Lemma 3 ......................................................................................................... 37

Proof to Proposition 1 ................................................................................................... 38

Proof to Proposition 2 ................................................................................................... 39

Proof to Corollary 1 ...................................................................................................... 40



3

Acknowledgement

I would like to thank my thesis advisors, Dr. Rachel Kranton and Dr. Michelle Connolly,

who have given me invaluable guidance. This piece of work would not be possible

without their advice and encouragement. My heartfelt gratitude goes out to my peers

from the thesis workshop, Jennifer Garand and Suhani Jalota, who made the arduous

process of thesis writing enjoyable. Our discussions have shaped my thinking and the

direction of this research. I would also like to thank Dr. Philipp Sadowski, Kaiyu Lee,

Jiakun Xu and Renhao Tan for bouncing ideas and offering feedback.

4

Abstract

This paper presents an economic model of the effects of identity and social norms on

consumption patterns. By incorporating qualitative studies in psychology and sociology, I

propose a utility function that features two components – economic (functional) and

identity elements. This setup is extended to analyze a market comprising a continuum of

consumers, whose identity distribution along a spectrum of binary identities is described

by a Beta distribution. I also introduce the notion of salience in the context of identity and

consumption decisions. The key result of the model suggests that fundamental economic

parameters, such as price elasticity and market demand, can be altered by identity

elements. In addition, it predicts that firms in perfectly competitive markets may

associate their products with certain types of identities, in order to reduce product

substitutability and attain price-setting power.

JEL codes: D11, D21

Keywords: identity, consumption, firm theory, social norms, heterogeneous agents

5

On January 24th, Apple Computer will introduce Macintosh.

And you’ll see why 1984 won’t be like “1984”.1

—Apple Super Bowl Commercial, 1984

1. Introduction

Apple’s 1984 commercial marked the start of its aggressive campaign to define

the Apple computer as the hallmark of contrarian identity. The campaign characterized

Mac users as creative, revolutionary and non-conforming, as opposed to brainwashed

‘proles’ in George Orwell’s 1984 (Livingstone, 2011). Following “1984”, Apple’s

subsequent advertisements, such as “Think Different” and “Get a Mac” in 2007, were so

successful that it gave rise to a brand community that Belk and Tumbat (2005) termed

“the cult of Macintosh.”

Olins (2000, as cited in Livingstone, 2011) attempted to provide an explanation

for the success of Apple’s campaigns: not only does branding alert consumers to the

existence of new products and signal product quality under asymmetric information, but

it also gives consumers the opportunity to define and express their identity. In other

words, Apple’s success in reshaping consumption behavior was predicated upon factors

beyond product functionality; it won the battle in consumer psychology and appealed to

the identities of its market population. In the short run, Apple positioned itself as a

contrarian product and appealed to consumers who adopt or wish to adopt contrarian

identities. In the long run, by describing conformity as Orwellian, Apple is pushing more

consumers to adopt contrarian identities, which it described as ‘creative’ and

‘revolutionary’.

Today, we observe a systematic difference in computer purchase patterns across

different populations. For example, while the total market share of Apple operating

systems is 5.08% in 2010 (Net Market Share, 2015), in the same year, 27% of college

student-owned laptops and 14% of college student-owned desktop computers are Macs

(Elmer-DeWitt, 2010). Theories exist to explain this disparity (e.g. different technical

specifications might make the different laptops useful for different purposes). However, it 1 Apple Inc (1984), 1984. https://www.youtube.com/watch?v=2zfqw8nhUwA.

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might also be possibly caused by identity and social norm – colleges are perceived to be

places of creativity and its students more likely to identify with traits that are associated

with the Apple computer. This hypothesis is supported by research in other social

sciences like consumer psychology. Oyserman (2009), for example, asserted that

identity-based motivation has rich implications for consumer behavior, observing that

“identity-congruent choices are more likely than identity-incongruent choices, no matter

if these choices are perceived as beneficial or not.”2

The market for Apples suggests that short and long-term consumption patterns are

shaped by forces beyond prices, quality and income – variables commonly used in the

formal economic analysis of consumption. More likely, there exists a host of

determinants for consumption behavior, including identity and social norms.

Incorporating identity considerations into economic analysis will be a step towards

explaining why consumption behavior shifts, beyond saying that they simply do, due to

some exogenous factors.

Indeed, econometrics have long relied on observable traits such as distribution of

identities and shifting demographic trends to predict economic outcomes. Yet, standard

economic theory has not fully taken into account this social dimension of choice, and

falls behind other social sciences in this respect. Akerlof and Kranton (2002) lamented

that economists have traditionally stayed away from these concepts, because they are

“too fuzzy to inform either theory or empirical work.”3 The current state is regrettable,

because “without a model that mirrors this sociology, economic analysis produces only

partial answers to key questions.”4 Indeed, that an economic concept as fundamental as

consumption choice fails to take into account important social dimensions diminishes and

limits the potential of economic analysis. This paper attempts to fill this gap.

Hence, this paper will present an economic model of the effects of identity and

social norms on consumption patterns. Using assumptions that are guided by qualitative

studies in psychology and sociology, I first develop a utility function for a consumer

choosing between two goods which are functional substitutes. This utility function 2 Oyserman (2009), p 257. 3 Akerlof and Kranton (2002), p 1168. 4 Ibid.

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features two components – economic (functional) and identity elements. I choose

functional substitutes to establish a ceteris paribus condition, so that identity effects may

be easily analyzed. I then extend this basic model to analyze a market comprising a

continuum of consumers, whose identity distribution along a spectrum of binary identities

(e.g. gender, race 5 , sexual orientation) is described by a Beta distribution. Having

produced an analytical form for market demand and relative price elasticity6, I proceed to

analyze how the salience 7 of an identity to a consumption decision can shape

consumption patterns. I also discuss possible ways future research may model the

dynamics of shifting identities.

There are two key results. First, identity elements can result in consumers being

less sensitive to changes in prices. Furthermore, if identity effects are sufficiently large,

there exists parameters for which an individual’s consumption decision ceases to be

influenced by price. While identity effects can shape the price elasticity of demand, the

exact relationship is dependent on two parameters – distribution of identity and prevailing

prices. For the case where relative prices are reflective of the relative marginal functional

utility of the goods and the distribution of identity is symmetric, an interesting result

arises. In Scenario 1, where the population is polarized in their identities (i.e. a majority

of consumers adopt extreme identities), both goods tend to be relative price inelastic. In

Scenario 2, where the population is neutral in their identities (i.e. a majority of consumers

has moderate identities), both goods tend to be relative price elastic. In the extreme case

where every consumer has the neutral identity, the problem resolves into one where there

are no identity effects. We know from standard economic models of substitutes that

goods are perfectly price elastic under this circumstance.

5 In this analysis, we will simplify race into two categories, e.g. Black and White. In reality, race is a complex issue. Section 5 offers speculation on how we may deal with multiple dimensions of identities. 6 Because this analysis relies on the relative price between two substitutes, the price elasticity function necessarily contains the possibility for substitution. Relative price elasticity may be interpreted as a hybrid of price elasticity and cross price elasticity. 7 Salience, as in behavioral economics, is the relevance of a certain consideration to a decision. To provide an example in the context of identity, we see that gender could be a relevant consideration when an individual decides whether to buy a dress or pants. But in other consumption decisions, such as whether to purchase a cup of coffee or tea, gender could become irrelevant.

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Second, if third parties can shape the salience of an identity to the consumption of

a good, they can, in doing so, alter the relative price elasticity of the good. Furthermore,

for the special case, where the identity distribution is symmetric and relative prices

reflective of marginal functional utility, the limit to which third parties can influence the

price elasticity is greatest for the case when the population is polarized.

These results have implications for firm theory and public policy. Identity

elements open up a new dimension in which firms may compete to maximize their

profits. This model predicts that firms whose products are easily substitutable may

engage in strategies to associate their products with a polarized identity, in order to

generate relative price inelasticity through identity effects and thereby maximize profits.

For example, Coca-Cola, whose competitors produce substitutable goods (soft drinks),

would want to associate their product with certain sports teams or celebrities so as to

appeal to individuals who identify as fans. In addition, it is conceivable that policymakers

would like to shape certain consumption patterns, e.g. discourage consumption of

unhealthy food, or encourage uptake of healthcare policies. Appealing to identity

elements could become a policy tool for such purposes.

The potential for such applications suggest that there are compelling reasons for

wanting to understand the interaction between identity and consumption. Section 2

provides a review of studies in psychology and identity economics necessary to form

assumptions about this interaction. In Section 3, I introduce an identity-utility model for a

single consumer choosing from two functionally substitutable goods that have different

identity associations. In Section 4, I extend this basic model to a multiple-consumer case,

in order to understand the interaction between market demand, price and identity. In

Section 5, I introduce the concept of salience with respect to identity. Section 6 outlines a

worked example in which this model may be applied. Section 7 concludes.

2. Literature Review

The work of incorporating social dimensions into economics was first taken up by

Akerlof and Kranton (2000), which formalizes identity economics. The idea that

underscores identity economics is that social groups – even arbitrarily constructed ones –

can impact behavior and economic outcome. Akerlof and Kranton (2000) constructed a

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neoclassical model 8 that incorporates identity-motivated behavior. Specifically, their

model relies on four key observations documented by sociology and psychology: (1) an

individual’s own actions give rise to identity-based payoffs, (2) actions have identity-

based externalities on others, (3) third parties can shape an individual’s payoffs, and (4)

whether a person has choice in her own identity varies across persons and situations.

Their model successfully explained phenomenon unaccounted for by standard economic

theory, such as gender-based occupational segregation in the workplace, and ‘self-

destructive’ behavior among the socially marginalized.

Since Akerlof and Kranton (2000), other studies have refined and extended the

model to examine education policies (Akerlof & Kranton, 2002), review principle-agent

and economic organization (Akerlof & Kranton, 2005), explain income inequality in

China (Afridi, Li, & Ren, 2015), and account for behavioral economic observations such

as altruism (Chen & Li, 2009). To the best of my knowledge, Sexton and Sexton (2014)

is the only study to provide a formal treatment on how identity and social norms may

affect consumption behavior. Sexton and Sexton (2014) restricts their analysis to the

consumption of green products. According to their theory, consumers would be willing to

pay more for green products if their purchases were observable and rewarded with

affirmation by other environmentally conscious individuals. This assumption deviates

from other studies in identity economics, which emphasize the intrinsicality of identity-

derived utility.

While research on the connection between identity and consumption has been

limited in the field of economics, other social sciences, such as consumer psychology

have found evidence on the connection through marketing data (Lamont & Molnár,

8 Akerlof and Kranton (2000) modelled identity-motivated behavior in the following manner. They construct the following utility function an individual j. !" = !"(&", &(", !)") We can interpret this equation in the following way: the utility of an individual j depends on the economic consequence of his own actions &", and on others’ actions &(". In addition, utility depends on j’s identity utility !)", which is in turn represented as: !)" = !)"(&", &("; ,", ε", .) This means that identity-derived utility is dependent on j’s own actions and others’ actions, given j’s assignment of people to social categories ,", the prescribed actions for each social category ., and her own characteristics ε".

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2001). The psychology literature in identity and consumption that I surveyed is founded

on Higgins (1987)’s work on self-discrepancy theory. In simplified terms, the self-

discrepancy theory states that individuals want to engage in behavior that is consistent

with their personal and social identities, failing which they could experience negative

feelings such as guilt and shame. My work will focus on social identity, which is how we

view ourselves in the context of our social groups and relationships.

Self-discrepancy theory gives rise to three key insights about identity and

consumption behavior.

Observation 1 Consumers prefer goods that “express and define (their) group

membership through shared consumption symbols.”9

Observation 2 Consumption decisions become reflexive once they are

established around an identity (Oyserman, 2009). Higgins

(2005) suggests that this phenomenon stems from a “feel right”

experience, known in psychology as regulatory fit.

Observation 3 Consumers prefer goods (or brands) that align with their

identity, but only when this identity is made salient (Aaker,

1999; Loken, 2005; Oyserman, 2009).

In the rest of the paper, I will present a consumption model that is consistent with

these three observations. In the next section, we begin with a basic model that considers a

one-consumer, multiple good scenario.

3. Basic Model

In order to isolate the effects of identity, we will focus our analysis on goods that

have equal functionalities. First, let us consider a standard neoclassical model, describing

three goods X, Y and Z. X and Y are perfect substitutes in terms of functionality, e.g. two

different brands of computers of exactly the same quality, while Z is a composite good,

9 Boorstin (1973), as quoted in Belk (1988), p 152.

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representing all other consumption. We can specify a Cobb-Douglas economic utility

function for an individual, /, that is a function of her consumption of X, Y and Z:

!0" = 1 + 3 456(4 (1)

where 7 ∈ 0, 1

Let ;" be the income for individual /. To simplify the problem, we can normalize the price

of Y to 1. We can then interpret <=, as the ratio of the price of X to the price of Y. Hence,

individual / will have the following budget constraint:

<=1 + 3 + <>5 ≤ ;" (2)

Solving this utility maximization problem, we find that individual / would consume only

good X if the ratio of price of X to that of Y is less than or equal to their relative marginal

utilities, and only good Y otherwise. We conclude that using standard microeconomic

models, for a specified utility function, the relative prices of goods X and Y are the sole

determinants of her consumption pattern – a notion that is already intuitive to us.

In the next step, we introduce identity-based motivation into the model, and

examine the implications. Let us define two social categories @ = AB, AC , with

prescribed consumption D, E respectively. Let F be a variable such that F ∈ −1, 1 . For

an individual /, F" > 0implies that individual / identifies with social category AC, while

the converse is true. Also, F" = 0if and only if individual / identifies with neither social

category. This specification allows us to interpret F" as the strength with which

individual / identifies with her social category. To give an illustration, say AB = IJKL

and AC = MLIJKL. If an individual has F" = −0.5, he identifies with being male with a

strength of 0.5.

The studies presented in Section 2 demonstrate that an individual gains utility by

consuming her prescribed good, and suffers utility loss for doing otherwise. We

incorporate Observation 1 into our model in following way. In addition to the economic

utility that the functionality of good X and Y confers, they also produce identity utility

(!)"). We can represent individual /’s identity utility as follows:

!)" =F"F"∙ 1 − 3 (3)

where F" ≠ 0

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Note that RSRS

is an indicator function. For example, when F" < 0 , RSRS= −1 ⇒ !)" =

−1 + 3. Hence, individual / obtains positive utility for consuming her prescribed good

(good Y in this case) and negative utility for consuming good X. For cases where F" = 0,

we intuitively conclude that there is no identity utility conferred, i.e. !)" = 0. These

outcomes are consistent with Observation 1.

To obtain the total utility, we can introduce a weight to the economic and identity

utility. Let the weight of identity utility in total utility be represented by the function

V F" , that is strictly increasing on identity strength. Consequently, the weight on the

economic utility is 1 − V(F").10 One possible function for V F" is:

V(F") = F"W (4)

Notice that this function satisfies VX F" ≥ 0, and 0 ≤ V F" ≤ 1 for domain of F" .

Hence, we obtain the total utility function for individual /:

!" = 1 − V(F") 1 + 3 +

F"F"∙ V F" 1 − 3

4

5 6(4 (5)

In order to solve the maximization problem for individual j, we first make a simplifying

assumption.

Assumption 1 Assume only corner solutions for X and Y. That is, individuals will

never choose to consume both X and Y. Formally, 1"∗ ∙ 3"∗ = 0.

We will postpone the justification for Assumption 1 to Section 4, where it will

become clear why it holds in a market with a continuum of consumers. Given

Assumption 1, we can derive Lemma 1.

10 Notice that the model could have simply used F" instead of V(F") to reflect the weight in the utility function, since the degree of an identity strength may be arbitrarily determined, as long as it is monotonic. However, I decide to introduce a separate mapping function, to allow for the possibility that F" is an empirically obtained variable that does not vary proportionately with its weightage in the utility function. In this case, a mapping function is useful.

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Lemma 1 Individual / will devote 7 of her income to either good X or Y, under

utility maximizing conditions:

<=1 + 3 = 7;" (6)

Proof

Found in Appendix ∎

Lemma 1 tells us that expenditure on good X and Y is independent of composite good Z.

This leads us to Lemma 2, which simplifies the maximizing problem.

Lemma 2 Solving the following problem maximizes individual’s /’s utility:

max=,_

!4"

where!4" = 1 − V(F") 1 + 3 +F"F"∙ V F" 1 − 3

(7)

subject to constraint <=1 + 3 ≤ 7;"

Proof Found in Appendix ∎

Proposition 1 The Marshallian demand for an individual / is described by the

following equation:

1"∗ <=, F", ;" =

7;"<=, <= <

1 − V F" +F"F"

V F"

1 − V F" −F"F"

V F"

7;"<=, 1 − V Fd −

F"F"

V F" < 0

0,otherwise

(8)


14

From Proposition 1, we observe that identity effects cause demand to become less

price sensitive. More accurately, a consumer who identifies more strongly with identity

AC will face greater inertia in substituting Good E away for Good D in response to

increase in relative price of GoodE. This result is consistent with Observation 2, that

consumption decisions become reflexive after they are established around an identity.

Furthermore, Proposition 1 suggests that it is possible to obtain complete price

inelasticity in the presence of identity effects. More specifically, there exists a threshold

or critical identity strength above which an individual’s consumption decision between

goods X and Y becomes independent of price, if price cannot take negative values. Above

the threshold identity strength, the individual will consume only her socially prescribed

good, even if she derives identical non-zero economic utility from consuming either

good. Corollary 1 is a formal statement of this result.

Corollary 1 ThereexistssomeV F" where 1 − V F" > 0 such that the

optimal consumption of Good X, 1"∗ = 0∀<= ≥ 0.


Standard models suggest that consumption decisions between two functionally

substitutable goods are contingent on relative price. The identity model, however, allows

for the possibility that price becomes an irrelevant factor in a consumer’s consideration,

should social pressure against consuming the good be sufficiently large. The model also

provides an explanation for phenomenon such as price disparity between two

economically identical products, and might even account for behavior such as boycotts. 11

11 To help the reader visualize how this phenomenon might take form, I offer a brief but prominent example: the boycott of sugar in England during the 1800s. As the anti-slave trade movement gained momentum, a growing number of people began to identify strongly with the movement. Sugar, viewed as the cruel product of slavery, became objectionable to individuals in this social category (Hochschild, 2006). In terms of the model, we can think of the anti-slavery identity as Ak, and sugar as good Y. For these individuals, V F" is sufficiently high such that no price could convince them to consume sugar.

15

We looked at the case for a single consumer, and examined her consumption

behavior. The natural next step is to expand the one-consumer model, and examine the

implications for the many-consumer case.

4. Market Demand

Section 4 is divided into 4 sub-sections. Section 4.1 lays out the model

specifications and introduces the Beta distribution for modeling population identity

distribution, Section 4.2 provides an interpretation for the variables, Section 4.3 solves

the model and produces an analytic expression for the market demand for good X, and

Section 4.4 illustrates the interaction between identity and market demand.

4.1. Model Specification

We consider a market with a continuum of consumers, who have utility functions

described by Equation (5) – note that this does not assume that consumers have the same

identity. For convenience, we normalize the number of consumers to 1, and assume that

each consumer has income ;. We can then attempt to describe the distribution of identity

strength of consumers in the market. For that purpose, this paper has chosen to model

after a Beta distribution because of the following:

i. The Beta distribution is a bounded distribution (between 0 and 1). This

property allows for a simple transformation of a Beta distributed random

variable to identity strength, which takes on a bounded value between -1

and 1.

ii. The shape of the Beta distribution is variable. This property has two

advantages. First, we have a model that allows for a higher degree of

generality in terms of how identity strength is distributed. Hence, the

model is versatile and applicable under a wider range of circumstances.

Second, we can model shifts in identity distribution by altering the shape

parameters of the Beta distribution.

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Let Fl~Beta n, o , where n and o are shape parameters and n, o > 0 . The

probability distribution function (PDF) of Fl is defined by the Beta distribution function:

pqr Fl; n, o = M(Fl; n, o) =Fls(6 1 − Fl

t(6

u(n, o) (9)

where u n, o = Fls(6 1 − Fl

t(66v wF"

We adopt the following notation for the cumulative distribution function (CDF) of Fl:

xqr Fl; n, o = r Fv; n, o = M Fl; n, o wFlRy

v (10)

In order to map the Beta distributed random variable (Fl) to identity strength F",

consider z: Fl → F" , a change of variable transformation described by z Fl = 2Fl − 1.

By inspection, z is an isomorphic and monotonic transformation. Consequently, we can

derive an inverse function z(6: F" → Fl , given by z(6 F" =6kRSW

. Applying

transformation z to Fl , we obtain a transformed Beta distribution of identity strength

across consumers in the market.

4.2. Interpretation of Shape Variables

In this subsection, I give an interpretation for the shape variables n and o. We

may interpret 6s

as the pressure on the population identity distribution towards A( .

Ceteris paribus, a higher 6s represents a higher proportion of the population adopting the

AB identity with greater strength. Conversely, 6t

represents the pressure towards AC. For

illustration consider the following scenarios represented in Figure 1.

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Figure 1 Scenarios illustrating the interpretation of pressure

Scenarios

Low pressure to both High pressure to AB Low pressure toAC

High pressure to both

Values of ~ and �

1n< 1;

1o< 1

or

n > 1; o > 1

1n> 1;

1o< 1

or

n < 1; o > 1

1n> 1;

1o> 1

or

n < 1; o < 1

Distribution

Description

Identity distribution is flushed towards the middle. Few adopt strong identities, most are moderate.12

Identity distribution is flushed towards the left. A large proprotion of the population adopts strong AB identities.13

Identity distribution is flushed towards both sides. Population is polarized in terms of their identities.14

We will return to using this notion to describe the interaction between

consumption and the dynamics of identity. Prior to that, it is appropriate to solve the

model and derive an analytic form for the market demand curve, in order that we may

begin to understand the impact of identity.

12 An example is socioeconomic status. A relatively large proportion of the population would identify as middle class; of course, with rising inequality, the pressure towards both ends may be increasing, i.e. higher kurtosis. 13 For example, views on slavery in modern times, where AB represents anti-slavery, while AC represents pro-slavery 14 For example gender – a large majority of the population identifies as male or female in approximately equal proportions.

18

4.3. Solving the Model

Using these specifications, we obtain an expression for the market demand for

Good E, by summing up individual consumption demand across the market, that is:

E = 1"∗ F", <=, ;" (11)

Lemma 3 Market demand can be written as the expectation of 1∗, given the

distribution of consumer identity strength in the market:

E <=; n, o = Ä 1∗

= 1"∗ F", <=, ;" ∙ M z(6 F" ; n, o wF"6

(6

(12)


To solve Lemma 3, we consider the following approach. From Proposition 1, we

know that for a given identity strength Fv, there exists a price <=v Ry, above which the

consumer stops consuming good X. Conversely, she consumes 4ÅSÇÉ

units of good X and

zero units of good Y when <= < <=v Ry. We will thereafter refer to this price <=v for a

given Fv as the critical price. From Proposition 1, we can derive Equation (13), which

tells us the critical price for given − 6W≤ Fv <

6W.

<=v Ry=

11 − 2V Fv

, 0 ≤ Fv <12

1 − 2V Fv , −12≤ Fv < 0

(13)

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Figure 2 Critical price-identity strength curve defined by Equation (13). On the

line, consumers are indifferent between consuming good X and Y. Above the curve,

consumers only consume good Y; below the curve, consumers only consume good X.

Figure 2 illustrates Equation (13); it traces the set of <=v Ryfor each Fv, at which a

consumer is indifferent between good X and Y. Above the line, where <= > <=v Ry, the

consumer consumes only good Y. Conversely, the consumer consumes only good X

below the line, i.e. <= < <=v Ry. The unrepresented domains, Fv >

6W and Fv < − 6

W is a

consequence of Proposition 1 and Corollary 1, which tells us that price ceases to matter

beyond a certain threshold identity strength. For example, when an individual’s identity

strength is larger than 6W, she only consumes good X, as illustrated in Figure 2.15

From Figure 2, we note that the relationship between <=v and Fv is monotonic. To

express this relationship in a convenient manner, let us denote the mapping from Fv to

15 More accurately, the consumer might consume good Y if they were compensated to do so, i.e. the price ratio <=v Ry

< 0, denoting that the price of good Y is negative. In reality, negative prices can and have occurred. However, for simplicity, we will assume that prices cannot take on negative values.

1√2

−1√2

Consume only X

Consume only Y

<=vÖRy

Fv

20

<=v Ry as Ü: Fv → <=v Ry

, for Fv ∈ − 6W, 6W. For Fv ≥

6W

, we take <=v Ry= ∞ to

represent that the consumer will consume Good E regardless of price, i.e. there is no

effective threshold price, as noted in Result 1. For Fv < − 6W, <=v Ry

< 0, implying that

good X will not be consumed.

The duality to the above proposition is that, for a given finite and non-negative

<=v , there exists an identity strength Fv ÇÉy, above which a consumer consumes only

Good X. Hence, there exists an inverse function for Ü , such that Ü(6: <=v → Fv ÇÉy,

within the domain 0 ≤ <=v < ∞. We will thereafter refer to this Fv ÇÉy as the critical

identity strength.

At this point, we make a short digression, to revisit the justification for

Assumption 1. We note that for a given non-zero and finite <=v, consumers will only

consume both good X and Y if their identity strength coincides with the critical identity

strength, i.e. F" = Fv ÇÉy. Since F" follows a continuous distribution, the probability that F"

is exactly Fv ÇÉy, p F" = Fv ÇÉy

≈ 0 ⟹ ä RSãRy ≈ 0 , where ä RSãRy åÉy denotes the

number of consumers with identity strength exactly equal to Fv ÇÉy. Therefore, we can

assume, to first approximation, that no individual will consume both good X and Y.

We can derive the market demand for X by noting the following two observations:

i. For a given <=v, consumers with F" > Fv ÇÉy will consume only good X, while

the others will consume 0 units of good X. Hence, we can find the number of

consumers who consume good X (denoted as ä=) by finding the number of

consumers with identity strength above Fv ÇÉy (ä RSçRy åÉy

). Hence, using the

Beta distribution, we can obtain ä= as follows:

ä= = ä RSçRy åÉy= 1 − r z(6 Fv ÇÉy

; n, o (14)

We can rewrite Equation (14) in terms of a given price <=v, using the transformation Ü:

ä= <=v; n, o = 1 − r z(6 Ü(<=v) ; n, o (15)

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ii. Recall from Proposition 1 that, above the critical identity strength, consumers

will consume 4ÅÇÉ

units of good X, regardless of the degree by which their

identity strength F" exceeds the critical level.

Proposition 2 The solution to the market demand for Good X is:

E <=, ;; n, o, 7 = 1 − r z(6 Ü <= ; n, o ∙7;<=

(16)


Next, we would like to understand how characteristics of the market demand is

shaped by the identity pressures in the population.

4.4. Interaction between Identity and Market Demand

In the context of this problem, where we are understanding the effects of identity

and prices on the consumption decision between two products, it is perhaps most

important to consider the relative price elasticity under the various scenarios described in

Figure 1. Note also that <=is the relative price of Good E to D.

Proposition 3 The relative price elasticity for good X is given as:

éE <=, ;; n, o, 7é<=

= −ér F; n, o

éF∙éz(6 FéF

∙éÜ <=é<=

∙7;<=

− 1 − r z(6 Ü <= ; n, o ∙7;<=W

(17)

Proof By partial differentiation, noting that z(6 maps F to F, and Ü

maps a given<= to F. ∎

22

It follows from Proposition 3 that identity elements modify relative price

elasticity, but the exact outcome depends on the distribution of identity and the initial

relative price of Good E and D. Corollary 2 describes results for a special case, where the

initial relative price is equal to the relative marginal functional utilities of Good X and Y,

in this case, <= = 1, and the distribution of identity strength is symmetric.

Corollary 2 For the case of <= = 1, i.e. the ‘neutral’ price, Good E and D are

less price elastic if there is an equally strong identity pressure

towards both goods. Conversely, Good E and D are more price

elastic in a market with equally weak identity pressure towards both

identities. Formally,

éE <=, ;; nè, oè, 7é<=

<éE <=, ;; nê, oê, 7

é<=

where<= = 1, nè = oè > nê = oê

(18)


Corollary 2 builds on Corollary 1, by generalizing the effects of identity on price

sensitivity of demand to the market level. The ‘neutral’ case put forth in Corollary 2

offers us intuition about such effects. In a high pressure condition, the consideration

given by consumers to identity outweighs that to price. Hence, changes in price results in

a marginal change in quantity demanded of a good. Figure 3 provides a graphical

interpretation of this intuition.

23

Figure 3 Graphical representation of the price elasticity of Good X in low and high

pressure markets

Graph Description

Initial Condition

Initially, <= = 1. At this price level, the critical identity strength is neutral, i.e. Fv ÇÉã6 = 0. Consider a marginal increase in <= to <=,6. The corresponding critical identity strength increases to Fv ÇÉãÇÉ,ë so that Fv ÇÉãÇÉ,ë > 0.

Impact on High Pressure Market

At first, when Fv = 0, half of population has F" > 0 = Fv, hence half of population consume good X, at 4Å

ÇÉ unit per person.

After price increases, Fv (indicated by extended vertical line) shifts right to Fv > 0. The proportion of the consumers with F" > Fv decreases by the amount represented by the red shaded area. Change in consumption of Good X is represented by the product of the shaded area and 4Å

ÇÉ.

Impact on Low Pressure Market

Likewise, in the low pressure market, the proportion of consumers who consume Good X decreases, but by a far larger number given by the shaded area. Note that the decrease in consumption of Good X falls by a much greater amount.

<=

increases

Corresponding increase in Fv

24

5. Salience

At this point, we have developed an intuition and nature of the interaction

between identity and the characteristics of the market demand, specifically price elasticity

of demand. We also observed that starting from a neutral point, demand is inelastic in a

high pressure market, and elastic in a low pressure market. We immediately see

implications for firm strategy, among other phenomenon – in a perfectly competitive

market or oligopoly, firms might attempt to associate their product with identities that

have high pressure in order to reduce their price elasticity and therefore their ability to set

prices.16 This section introduces parameters that describes the association of a product

with an identity.

Oyserman (2009) asserts that multiple social identities can functionally add

together, such as being white and male. It is appropriate to isolate a set of identities,

insofar as it is salient, or “relevant in a particular context.” Under different cues and

situations, identities can increase in temporary relevance. McGuire, McGuire, Child and

Fujiota (1978, as quoted in Oyserman, 2009), for example, show that ethnicity becomes

salient when one’s ethnicity stands out in a social environment. In addition, consumers

prefer goods (or brands) that align with their identity, but only when this identity is made

salient (Aaker, 1999; Loken, 2005; Oyserman, 2009) – Observation 3. Hence, identity

can only affect consumption behavior if the identity in question is relevant or salient to

the consumption decision.17 For example, under ordinary circumstances, our choice of

16 For illustration, consider Coca-Cola, which sponsors sports events, such as the Olympics. Coca-Cola advertises itself as the drink for athletes (whether this is a recommended option from the health perspective is a separate question). We may think of the AC identity as ‘sporty’ and AB as otherwise. Having established itself as the prescribed good for the AB identity, Coca-Cola could transform its market from perfectly competitive (for soft drinks) to one where it has some pricing power. Future research may consider undertaking a survey of similar markets to verify whether this prediction is consistent with observations. 17 The notion of salience is not new in economics. In fact, some studies in identity economics such as Benjamin, Choi, and Strickland (2010) have employed salience to conduct laboratory experiments to verify the role of identity effects in economic behavior and outcomes. To study whether social norms associated with gender and race affect time and risk preferences, Benjamin et al. (2010) performed social “category-salience manipulations” using suggestive questionnaires, to prime gender and racial identity. It found that identity-related effects were significant when subjects were primed, but were insignificant otherwise. In response to this finding, Benjamin et al. (2010) offered the explanation that subjects who are primed conform to their social category’s social prescriptions, as Akerlof and Kranton (2000) and Higgins (1987) suggested.

25

laptop brands is independent of whether we adopt contrarian or conformist identities; we

may very well ignore the effects of identity on consumption. However, identity elements

become relevant after Apple successfully frames its product as a prescribed good for

contrarians.

To model this effect, we introduce a new variable, íC, such that 0 < íC ≤ 1, to

represent salience of an identity to the consumption decision of Good X. Let íC = 1 if the

identity in question is fully salient, and íC → 0 if it is irrelevant to the consumption

decision. To model the interaction between salience and pressure, we may introduce a

concept called salient pressure, denoted by 6síC and 6

tíC, such that the modified shape

parameters take on the following form:

nC =níC

(19)

oC =

oíC

(20)

Notice that salient pressure is increasing on salience and pressure. In the scenario where

an identity lacks salience in the decision in question, salient pressure takes on the value 0,

while the modified shape parameters nC, oC → ∞. Under this condition, the PDF of the

Beta distribution is such that the entire population has identity F = 0. Mathematically,

this simplifies into the neoclassical microeconomic model involving perfect substitutes,

where above a certain <C, consumers demand only Good Y, and below the certain <C,

only Good X. This is consistent with the notion that standard models should hold in the

absence of identity elements.

26

Proposition 4 Taking into account salience effects, we would like to re-define the

solution to the market demand as:

E <=, ;; n=, oC, 7 = 1 − r z(6 Ü <= ; n=, oC ∙7;<=

(21)

Also, the relative price elasticity for Good X may be re-defined as:

éE <=, ;; nC, oC, 7é<=

= −ér F; nC, oC

éF∙éz(6 FéF

∙éÜ <=é<=

∙7;<=

− 1 − r z(6 Ü <= ; nC, oC ∙7;<=W

(22)

Corollary 3 If third parties can shape the salience of an identity C to the

consumption of a Good X (and its functional substitute Good Y),

they can shape the relative price elasticities of Good X and Y.

Furthermore, in the ‘neutral’ case (as presented in Corollary 2), the

limit to which third parties can cause Good X and Y to be price

inelastic increases with the pressure of @ , i.e. when as n = o

decreases. Mathematically, we have:

minîï

éE <=, ;; n=è, o=è, 7é<=

< minîï

éE <=, ;; n=ê , o=ê , 7é<=

wheren=) =n)

íC, o=) =

o)

íC, ñ = {K, ℎ}

and <= = 1, nè = oè > nê = oê

(23)


Corollary 3 reflects Observation 3, that identity effects are present only if the

identity in question is salient to the consumption of Good X. Corollary 3 also reflects the

intuition that we stated in the introduction of Section 5. Assuming that a firm would like

to gain price-setting power by decreasing the price elasticity of its product, it might

27

consider anchoring its product in an identity of high pressure through the means such as

advertisement. Hence, our model predicts that firms with highly substitutable products

may engage in such advertisements, if the costs of doing so outweighs the additional

revenue resulting from its newly attained price-setting ability. Future research may verify

if this prediction is consistent with observations. And if the prediction is correct,

Corollary 3 may allow us to gain new understandings about firm strategy.

In addition, the notion of salient pressure also allows us to model a world with n

dimensions of identities, where each dimension is salient to different extents in different

consumption decisions. Instead of a one-dimensional Beta distribution, we would obtain

a multi-dimensional joint probability distribution, weighted by the salience of the choice

in each dimension of identity. However, in order to focus the discussion on the dynamics

of identity and its relation to consumption patterns, we will leave the generalization to

multiple dimensions of identities to future studies. Further studies might also examine the

decay of salience over time.

6. Chick-fil-A

In this section, I employ a worked example to illustrate the potential applications

of this model in firm strategy.

Consider the chicken sandwich market. Let be good X represent Chick-fil-A

chicken sandwiches, and Y represent chicken sandwiches from a comparable competitor.

For simplicity, we take the ratio of the price of Chick-fil-A to Brand Y chicken

sandwiches to be given and equal to 1, i.e. <= = 1. As in Section 4, consider a continuum

of consumers with a utility function described by Equation (5). Let AB denote the social

category of people who are pro-gay rights and AC, those who are anti-gay rights. The

distribution of the identity strength of the consumer population is given by a transformed

Beta distribution with shape parameters n and o.

At first, whether one is pro or anti-gay rights is irrelevant to the consumption

decision of chicken sandwiches. Therefore, íC → 0 , and the corresponding salient

pressure is 0. The market equilibrium problem simplifies into a standard neoclassical

problem. For simplicity, let us assume that market share between Chick-fil-A and Brand

Y chicken sandwiches is evenly split.

28

However, in 2012, Chick-fil-A announced that it opposes gay rights (Horovitz,

2012). Standard economic analysis does not predict any changes in the market share of

Chick-fil-A and Brand Y sandwiches, because one’s taste and preference for chicken

sandwich is functionally independent of one’s views on gay rights. Moreover, even if we

were to say that market share changes due to changing taste and preferences, we are

unable to comment on the direction of the change in market share. Instead, we can

analyze the situation using the identity model.

After the announcement using the model, one’s views on gay rights become

relevant to the consumption decision of chicken sandwiches. We may represent the event

as an increase of íC from 0 to 1. Chick-fil-A became associated with the anti-gay rights

identity, and for simplicity, we will take Brand Y as being associated with the pro-gay

rights identity. In this way, Chick-fil-A and Brand Y sandwiches become prescribed

goods for their respective social groups. As a result, salient pressure becomes non-zero.

From Proposition 1 and Corollary 2, we expect that consumers with F" < − 6W

will stop consuming Chick-fil-A sandwiches, regardless of <= . This prediction is

consistent with observations; after the announcement, “many gay rights groups called for

boycotts” of Chick-fil-A sandwiches.18

The next question we can ask is, how is the market demand for Chick-fil-A and

Brand Y sandwiches affected? The natural instinct is to jump to the conclusion that the

market demand for Chick-fil-A sandwiches decreased, given bad publicity. However, the

opposite happened; after the announcement, “consumer use of the chain was up by 2.2%”

while “market share was up by 0.6%.”19, 20

This apparent paradox is in fact resolvable. We note that Chick-fil-A operates

largely in southern states, where a more conservative stance towards gay rights is in

18 Horovitz (2012). 19 Ibid. 20 The disparity between the increase in consumer use and market share might be explained by new entry of consumers into the chicken sandwich market. It is conceivable that consumers may begin consuming previously unconsumed goods to support their identity and beliefs. More robust models may take into account this possibility. One possible model would allow 7 in Equation (1) to depend on identity strength, to reflect how consumers might shift some proportion of their income from other goods to the good in question – chicken sandwiches in this case.

29

general adopted. Hence, we would expect more individuals to belong to AC (anti-gay

rights) than AB (pro-gay rights). To model this observation in the Beta distribution, we

choose values for the shape parameters, such that n > o, i.e. 6s< 6

t, meaning that the

salient pressure towards anti-gay rights (6tíC) is larger that than towards pro-gay rights

(6síC). Let us pick n = 3, o = 2, in order to demonstrate using a concrete example. Since

we have assumed, for simplicity, that íC = 1 , the modified shape parameter (taking

salience into consideration) equals the value of the unmodified shape parameter. The

PDF of the transformed Beta distribution of salient identity strengths is illustrated in

Figure 4b.

Given that <= = 1 , we obtain the critical identity strength by taking the

transformation Ü(6 <= – as described by Equation (13) – as Fv ÇÉy= 0 . We apply

Proposition 2, and obtain that the quantity demanded for Chick-fil-A sandwiches is E∗ =

0.688 ∙ 7; (refer to Graph 4 for a graphical illustration). By symmetry, we can obtain the

demand for Brand Y sandwiches as D∗ = 0.312 ∙ 7; . The result is consistent with

observations that the demand and market share for Chick-fil-A sandwiches increased.

Recently, Chick-fil-A is “recalibrating its moral and culinary compass” as it

intends to move into more liberal cities, such as “Chicago, New York and Los Angeles,”

where we expect individuals to be friendlier to the gay rights movement. 21 This

phenomenon is consistent with Corollary 3. In addition, it is likely that the pressure

towards pro-gay rights (6t

) will increase as society becomes more progressive. Short of

flipping its position on gay rights, Chick-fil-A’s interest is to decouple itself from the

issue, i.e. reducing the salience of the issue to the consumption of its products.

21 Horovitz (2014)

30

Figure 4a Critical price-identity strength curve. Relative price <= is taken as given

and the critical identity strength Fv ÇÉ is in turn conditional on <=. In this example, <= =

1, which corresponds to Fv ÇÉ = 0 on the curve. The dotted line from Figure 4a through

Figure 4b indicates the critical identity strength.

Figure 4b The probability distribution function of F" follows a transformed Beta

distribution with shape parameters n = 3, o = 2 . Fv ÇÉã6 = 0 corresponds to an area

under the graph of 0.312, or xqr = 0.312 = äB, the number of consumers who only

consume Brand Y sandwiches. From Proposition 2, äC = 1 − xqr = 0.688, and the

market demand for Chick-fil-A sandwiches is E∗ = 0.688 ∙ 7;.

F"

pqr(F"; n = 3, o = 2)

<=

Fv|ÇÉ

Fv|ÇÉã6 = 0

Critical price-identity

strength curve

P.D.F of population

identity distribution

31

7. Conclusion

Studies in consumer psychology and sociology have, for a long time, recognized

the role of identity and social norms in consumption patterns. In this paper, I attempt to

incorporate these observations into an economic model that describes their effects on

consumption pattern.

The key result of the model suggests that fundamental economic parameters, such

as price elasticity and market demand, can be profoundly altered by identity elements.

We note in Section 5 and 6 that such an effect could have potential applications in firm

theory. For example, firms may compete on the identity dimension to maximize their

profits. This model predicts that firms in perfectly competitive markets may associate

their products with high-pressure identities, in order to reduce product substitutability and

attain price-setting power. Future work could use empirical data on observable identity

distribution, relative price and market share to estimate the values of nC, oC in different

markets. Identity elements also offer a new approach to understanding how firms such as

Apple optimize their level of advertising and their impact on the consumption of other

goods. It is also conceivable that policymakers are interested in influencing the level of

consumptions of certain goods, e.g. discouraging unhealthy food. Hence, understanding

the effects of identity on consumption patterns could open up a new set of policy tools.

In this model, we assumed identity to be a static construct. But over time, third

parties and changing social contexts can cause shifts in a person’s social identity

(Oyserman, 2009). Marketing campaigns, such as Apple’s, are immediate examples of

how third parties are able to and have the incentives to reshape identities in the long run.

And there exist a host of examples where social norms can shift, including changing

perception of gender roles, attitude towards environmental conservation, and tolerance of

alternative lifestyles. In addition, changing fertility rates can, and have caused shifting

demographics. Future research may consider extending this model to account for shifting

identity distributions. Such an extension would allow us to predict changes in

consumption pattern over time, which could be of value to firms and policymakers.

Economics is the study of choice. And the choices we make are inevitably

influenced by our beliefs, our community – our identity. The use of certain observable

identity distributions to predict and explain consumption decisions promises to expand

32

the scope of economic analysis. This paper presents a consumption model that attempts to

robustly account for such social dimensions. While this model remains incomplete amidst

the complications and nuances of human interaction, it is nonetheless a step towards

unravelling the complexity of our choices.

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35

Appendix

Proof to Lemma 1

Lemma 1 Individual / will devote 7 of her income to either good X or Y, under

utility maximizing conditions:

<=1 + 3 = 7;"

Proof Using Assumption 1, that only corner solution exists, say that

individual / only consumes good X, i.e. 3 = 0. We can rewrite utility

function of individual / as:

!" = 1 1 − V F" +

F"F"∙ V F"

4

5 6(4

Consider the case where F" ≥ 0, then the function simplifies to:

!" = 1456(4

which is a simple Cobb-Douglas specification, from which we know

<=1 = 7;". And for cases where F" < 0, the utility function simplifies

to:

!" = 1 − 2V F"41456(4

Again, we have a simple Cobb-Douglas specification, and we

conclude that <=1 = 7;", when 3 = 0. We can hence also write that

as <=1 + 3 = 7;" . By symmetry, we can show the same for when

1 = 0. Hence, Lemma 1 holds for all corner solutions of X and Y. ∎

36

Proof to Lemma 2

Lemma 2 Solving the following problem maximizes individual’s /’s utility:

max=,_

!4"

where!4" = 1 − V(F") 1 + 3 +F"F"∙ V F" 1 − 3

subject to constraint <=1 + 3 ≤ 7 ∙ ;

Proof The corollary to Lemma 1 is that individual / will devote (1 − 7);"

to consuming good Z, hence,

5∗ =

(1 − 7);"<>

We note that 5∗ ≥ 0 ⇒ 56(4 ≥ 0 . Rewriting the utility function

under optimizing conditions, we obtain:

!" = !4"45∗6(4

Taking partial differential of !" with respect to !4":

é!"é!4"

= 7!4"4(65∗6(4

If we can show that !4" ≥ 0 at the optimizing points, then ûüSûü†S

≥ 0.

We know that 1 − V(F") 1 + 3 ≥ 0. We note in the second term

that the consumer can always choose between 1 and 3, such that

term takes on a non-negative value. Hence, under optimizing

conditions, maximizing !4" will maximize !". ∎

37

Proof to Lemma 3

Lemma 3 The market demand can be written as the expectation of 1∗, given

the distribution of consumer identity strength in the market:

E(<=; n, o) = Ä 1∗ = 1"∗ F", <=, ;" ∙ M z(6 F" ; n, o wF"

6

(6

Proof Having normalized number of consumers N to 1, we can use the

convenient property that the area under the Beta distribution is also

1. We can then interpret the PDF, M F; n, o = M z(6 F ; n, o , as

the number of consumers with identity strength F. By this construct,

the market demand for good X becomes synonymous with the

expectation of 1∗ across the Beta function. ∎

38

Proof to Proposition 1

Proposition 1 The Marshallian demand for an individual / is described by the

following equation:

1"∗ <=, F", ;" =

7;"<=, <= <

1 − V F" +F"F"

V F"

1 − V F" −F"F"

V F"

7;"<=, 1 − V F" −

F"F"

V F" < 0

0,otherwise

Proof From Lemma 2, we can obtain the ratio of marginal utility of

X to marginal utility of Y as:

°!C°!B

=é!4"/é1"é!4"/é3"

=1 − V F" +

F"F"

V F"

1 − V F" −F"F"

V F"

Notice that the numerator and denominator cannot be negative at the

same time.

Case 1: both numerator and denominator are positive. Consumer

consumes only good X if £üï£ü§

> <=.

Case 2: numerator is positive and denominator is negative. Since

°!C > °!B, consumer only consumes good X.

Case 3: numerator is negative and denominator is positive. Since

°!C < °!B, consumer consumes zero unit of good X. ∎

39

Proof to Proposition 2

Proposition 2 The market demand can be written as

E <=, ;; n, o, 7 = 1 − r z(6 Ü <= ; n, o ∙7;<=

Proof From Lemma 3, we have:

E(<=; n, o) = Ä 1∗ = 1"∗ F", <=, ;" ∙ M z(6 F" ; n, o wF"

6

(6

Considering the integral piecewise gives us:

E <=, ;; n, o = 1∗ F", <=, ;" ∙ M z(6 F" ; n, o wF"

Ry åÉ

(6

+ 1∗ F", <=, ;" ∙ M z(6 F" ; n, o wF"6

Ry åÉ

where Fv ÇÉ is the critical identity strength for <= , above which a

consumer consumes only good X. From Equation (13), we have

1∗ F", <=, ;" (6•RS¶Ry åÉ= 0 and 1∗ F", <=, ;" Ry åÉ¶RS•6

= 4ÅÇÉ

. Then,

we can write the piecewise integral:

E <=, ;; n, o, 7

= 0 ∙ M z(6 F" ; n, o wF"Ry åÉ

(6

+7;<=∙ M z(6 F" ; n, o wF"

6

Ry åÉ

Simplifying, we obtain:

E <=, ;; n, o, 7 =

7;<=∙ M z(6 F" ; n, o wF"

6

Ry åÉ

We recognize that M z(6 F" ; n, o wF"6Ry åÉ

is the probability that

Fv ÇÉ < F" ≤ 1, which is equivalent to 1 − xqr(Fv ÇÉ). We can thus

rewrite the market demand for good X as:

E <=, ;; n, o, 7 = 1 − r z(6 Ü <= ; n, o ∙7;<=∎

40

Proof to Corollary 1

Corollary 1 ThereexistssomeV F" where 1 − V F" > 0 such that the

optimal consumption of Good X, 1"∗ = 0∀<= ≥ 0.

Proof Consider F" < 0 , that is individual / is prescribed Y. From

Proposition 1, rewrite condition for1)∗ = 0 as

<= > 1 − 2V F"

⇒1"∗ = 0∀<= ≥ 0 when V F > 0.5 , or 1 − V F" > 0 . By

symmetry, the same can be shown for Y. ∎

41


Corollary 2 For the case of <= = 1, i.e. the ‘neutral’ price, Good E and D are

less price elastic if there is an equally strong identity pressure

towards both goods. Conversely, Good E and D are more price

elastic in a market with equally weak identity pressure towards both

identities. Formally,

éE <=, ;; nè, oè, 7é<=

<éE <=, ;; nê, oê, 7

é<=

where<= = 1, nè = nê, oè > nê = oê

Proof From Proposition 3, we have

éE <=, ;; n, o, 7é<=

= −ér F; n, o

éF∙éz(6 FéF

∙éÜ <=é<=

∙7;<=

− 1 − r z(6 Ü <= ; n, o ∙7;<=W

For<= = 1, r z(6 Ü <= ; n, o = 0.5∀n = o . Hence, the only

term that differs is ûß R;s,tûR

.Recall that ûß R;s,tûR

= M F; n, o .

Substitute F = z(6 Ü <= = 0.5. Then, we may write M F; n, o as

M 0.5; n, o = n =0.5W(s(6)

1 − 1W s(6w16v

Using numerical methods, we may evaluate that M 0.5; n, o = n is

strictly increasing in n for n > 0. Hence,

ér F; nè, oè

éF>ér F; nê, oê

éF

Then the price elasticity when n = nè , o = oè is more negative

than when n = nê,o = oê. Hence, when we take the absolute:

éE <=, ;; nè, oè, 7é<=

<éE <=, ;; nê, oê, 7

é<=∎

42


Corollary 3 If third parties can shape the salience of an identity C to the

consumption of a Good X (and its functional substitute Good Y),

they can shape the price elasticities of Good X and Y. Furthermore,

in the ‘neutral’ case (as presented in Corollary 2), the limit to which

third parties can cause Good X and Y to be price inelastic increases

with the pressure of @ , i.e. when as n = o decreases.

Mathematically, we have:

minîï

éE <=, ;; α=è, o=è, 7é<=

< minîï

éE <=, ;; n=ê , o=ê , 7é<=

wheren=) =n)

íC, o=) =

o)

íC, ñ = {K, ℎ}

and <= = 1, nè = oè > nê = oê

Proof From Corollary 2, we know that ûC ÇÉ,Å;sÉ© ,tÉ© ,4ûÇÉ

is minimized when

n=) = o=) is minimized. For a given n), n=) =s©

îï is minimized when

íC takes on the maximum possible value, i.e. íC = 1 . Hence,

minîï

n=) = n) . Therefore, minîï

ûC ÇÉ,Å;sÉ© ,tÉ© ,4ûÇÉ

= ûC ÇÉ,Å;s©,t©,4ûÇÉ

.

From Corollary 2, we know that ûC ÇÉ,Å;s™,t™,4ûÇÉ

< ûC ÇÉ,Å;s´,t´,4ûÇÉ

.

∎

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[Final] A Theory of Identity and Consumption